The fight over Social Security is, above all, about what kind of society we want to have. But it's also about numbers. And the numbers the privatizers use just don't add up.

Let me inflict some of those numbers on you. Sorry, but this is important.

Schemes for Social Security privatization, like the one described in the 2004 Economic Report of the President, invariably assume that investing in stocks will yield a high annual rate of return, 6.5 or 7 percent after inflation, for at least the next 75 years. Without that assumption, these schemes can't deliver on their promises. Yet a rate of return that high is mathematically impossible unless the economy grows much faster than anyone is now expecting.

To explain why, I need to talk about stock returns. The yield on a stock comes from two components: cash that the company pays out in the form of dividends and stock buybacks, and capital gains. Right now, if dividends and buybacks were the whole story, the rate of return on stocks would be only 3 percent.

To get a 6.5 percent rate of return, you need capital gains: if dividends yield 3 percent, stock prices have to rise 3.5 percent per year after inflation. That doesn't sound too unreasonable if you're thinking only a few years ahead.

But privatizers need that high rate of return for 75 years or more. And the economic assumptions underlying most projections for Social Security make that impossible.

The Social Security projections that say the trust fund will be exhausted by 2042 assume that economic growth will slow as baby boomers leave the work force. The actuaries predict that economic growth, which averaged 3.4 percent per year over the last 75 years, will average only 1.9 percent over the next 75 years.

In the long run, profits grow at the same rate as the economy. So to get that 6.5 percent rate of return, stock prices would have to keep rising faster than profits, decade after decade.

The price-earnings ratio - the value of a company's stock, divided by its profits - is widely used to assess whether a stock is overvalued or undervalued. Historically, that ratio averaged about 14. Today it's about 20. Where would it have to go to yield a 6.5 percent rate of return?

I asked Dean Baker, of the Center for Economic and Policy Research, to help me out with that calculation (there are some technical details I won't get into). Here's what we found: by 2050, the price-earnings ratio would have to rise to about 70. By 2060, it would have to be more than 100.

In other words, to believe in a privatization-friendly rate of return, you have to believe that half a century from now, the average stock will be priced like technology stocks at the height of the Internet bubble - and that stock prices will nonetheless keep on rising.

Social Security privatizers usually defend their bullishness by saying that stock investors earned high returns in the past. But stocks are much more expensive than they used to be, relative to corporate profits; that means lower dividends per dollar of share value. And economic growth is expected to be slower.

Which brings us to the privatizers' Catch-22.

They can rescue their happy vision for stock returns by claiming that the Social Security actuaries are vastly underestimating future economic growth. But in that case, we don't need to worry about Social Security's future: if the economy grows fast enough to generate a rate of return that makes privatization work, it will also yield a bonanza of payroll tax revenue that will keep the current system sound for generations to come.

Alternatively, privatizers can unhappily admit that future stock returns will be much lower than they have been claiming. But without those high returns, the arithmetic of their schemes collapses.

It really is that stark: any growth projection that would permit the stock returns the privatizers need to make their schemes work would put Social Security solidly in the black.

And I suspect that at least some privatizers know that. Mr. Baker has devised a test he calls "no economist left behind": he challenges economists to make a projection of economic growth, dividends and capital gains that will yield a 6.5 percent rate of return over 75 years. Not one economist who supports privatization has been willing to take the test.

But the offer still stands. Ladies and gentlemen, would you care to explain your position?